The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Cornelia Mihaila.
Fisher-KPP equations are a type of reaction-diffusion equations that model population dynamics. Their behavior is characterized by the invasion of an unstable state by a stable one, which leads to the phenomenon of spreading. In one dimension, it has been shown that localized data give rise to solutions that lag behind the slowest traveling front by a logarithmic term. In this talk, we will discuss the analogous periodic problem in R^n, where we determine the asymptotic rate of spreading of the solutions and the logarithmic correction along every unit direction.
A classical result of Wente, motivated by the study of sessile capillarity droplets, shows the axial symmetry of every hypersurface which meets a hyperplane at a constant angle and has mean curvature depending only on the distance from that hyperplane. We will prove an analogous result for the fractional mean curvature operator.
We consider a parameter estimation problem for finding the drift and volatility coefficient for a large class of parabolic Stochastic PDEs driven by space-time noise (white in time, and possible colored in space). In the first part of the talk, we focus on spectral approach and derive several classes of estimators based on the first N Fourier modes of a sample path observed continuously on a finite time interval. We will briefly review the maximum likelihood estimators, and then discuss a new class of estimators for the drift parameter, called the Trajectory Fitting Estimator (TFE). The TFE can be viewed as an analog to the least square estimator from the time series analysis. We will discuss consistency and asymptotic normality of such estimators as number of the Fourier modes tends to infinity. Some nonlinear SPDEs will be discussed too. In the second part of the talk we will discuss the parameter estimation problems for discretely sampled SPDEs. We will present some general results on derivation of consistent and asymptotically normal estimators based on computation of the p-variations of stochastic processes and their smooth perturbations, that consequently are conveniently applied to SPDEs. Both the drift and the volatility coefficients are estimated using two sampling schemes - observing the solution at a fixed time and on a discrete spatial grid, and at a fixed space point and at discrete time instances of a finite interval. The theoretical results will be illustrated via numerical examples.
Tissue growth, as in solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics. Several levels of mathematical descriptions are commonly used, including possibly elastic behaviours, visco-elastic laws, nutrients, active movement, surrounding tissue, vasculature remodeling and several other features. We will focuss on the links between two types of mathematical models. The `compressible' description describes the cell population density and a more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form. Including additional features also opens other questions as circumstances in which instabilities may develop.
In this work, we present new alternatives for the approximation of the homogenized matrix for diffusion problems with highly-oscillatory coefficients. These different approximations all rely on the use of an "embedded corrector problem", where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients are constant. The motivation for considering such embedded corrector problems is the fact that, for some particular and practically relevant heterogeneous materials (e.g. piecewise constant coefficients), they can be very efficiently solved. Several strategies are presented to appropriately determine the outside constant coefficients. We then prove that the different approximations we introduce all converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity. Some works in progress will also be mentioned. Joint works with E. Cancès, V. Ehrlacher, B. Stamm and S. Xiang. Organizer: CAMP/Nonlinear PDE
Radiation hydrodynamics is a widely used model in astrophysics simulations. In many situation, a diffusion approximation of the radiative part is sufficient to decribe correctly phenomena of interest. In this talk, I will review some existence results on such systems, with an additional damping term in the momentum equation and small data. It is based on the Kawashima-Shizuta method for hyperbolic-diffusion systems. Similar results are also proved for models including magnetic effects. Another widely used model is the compressible Euler model coupled to Poisson equation. Here, extending a method due to Grassin-Serre, we also prove global existence for this system, we prove existence of a global solution for small data. These are joint works with B. Ducomet (Univ. Créteil), R. Danchin (Univ. Créteil) and S. Necasova (Acad. Sc. Czech Republic, Praha)